The Compound Pendulum

The above equation can also be written including the mass of the pendulum. However, ‘at the same place, bodies of different masses fall in a vacuum with the same acceleration g’ 5 and, as mass is a constant, it can be ignored.

Prediction:

Squaring both sides of the above equation gives:

T² = 4π²  h² + k²

              g

Expanding the brackets gives:

T² = 4π² k² + 4π² h²

   hg          hg

Multiplying through by h gives the initial equation in the form of y = mx + c:

T²h = 4π² h² + 4π² k²

      g           g

Plotting T²h against h² will produce a straight line ‘of which the gradient gives 4π²/g’ 5. By rearrangement, the acceleration due to gravity can be found by:

g = 4π²

      m

where m is the gradient of the line. The acceleration due to gravity varies dependant upon altitude and latitude. At lower latitudes, the value for g will be smaller due to a centrifugal force acting perpendicular to the axis of rotation, which offsets the gravitational force on a body. Also, due to the bulge at the equator; objects are further from the core of the Earth and hence experience a smaller force than at other locations on the Earth. 9 

Using the International Gravity Formula 10, and taking into account that the latitude of Alnwick is 55.410307º 11, the value for gravity at the location which the experiment will take place is: 981542.3 mgals, which is 9.81 ms-2. A value close to this should be drawn from the gradient of the straight line graph.

The value for k is a constant, and consequently can be ignored, causing the equation to take on the form:

T²h = 4π² h²

      g          

This equation relates to that of the simple pendulum, the period of which can be found by 8:

T = 2π  h

             g

When rearranged into the form y = mx + c, it is clearly visible that the equation for the simple pendulum has been multiplied through by h to form the equation for the compound pendulum:

Preliminary Work:

A wooden metre rule drilled through its centre at 0.05m intervals was suspended freely on a pin after ensuring that the centre of gravity was at 0.5m along the metre rule. This was achieved by applying mass, in the form of blu-tack, to one end of the rule when it was suspended at 0.5m until it reached equilibrium and balanced - ‘The position of the centre of gravity can be determined with sufficient accuracy by balancing the pendulum’ 1 The force of gravity ‘can be considered as concentrated at the centre of gravity’ 6 so this must be found to determine h, and hence determine the value for g.  

A number of oscillations were timed for a range of values for h to decide upon an acceptable range of lengths and number of oscillations for the main experiment. Any possible errors were identified and will be explained in the following section, along with information on how the errors will be reduced in the main experiment.

• Previous to the preliminary, the possibility of discrepancies with the angle at which the pendulum was released was identified and combated. A piece of card was attached to the retort stand with a line ruled upon it at 40º to the vertical. The pendulum was held at rest upon the line before release to ensure that the angle at which it was released each time, and hence the height at which it was released, remained constant. (See Diagram A) However, it is highly difficult for an observer to stay constantly still for a large length of time. If the observation point is moved, even minimally, the results could be compromised and errors could be introduced due to parallax. For consistent results, the retort stand should line up with a fixed vertical line behind it. This would reduce the parallax errors and therefore increase the reliability of results.  

• The amplitude of each oscillation is not constant and reduces slightly each time. At the extremes of motion, the pendulum slows down and stops for a split-second as it changes direction. It is difficult to gauge at which point the pendulum chances direction, and - because of this - oscillations will be counted from the centre, where it passes the vertical line of the retort stand. Unlike at the extremes of motion, the pendulum will always pass this point, which improves the reliability of results. (See Diagram B).

• To prevent the retort stand moving unnecessarily, it was clamped securely to the desk using a g-clamp. Without the clamp the retort stand would sway with the oscillating pendulum hence affecting the results.
• To hold the pin firmly in place it must be pushed through a cork and clamped on the stand with the pin protruding horizontally. To ensure whether the pin is horizontal, it can be checked with a spirit level. The sharp end of the pin is dangerous if left exposed and should be covered with another cork. This also stops the metre rule from moving forward and slipping off the pin.
• Reaction times with starting and stopping the timer will be the greatest timing error. ‘The average human's reaction time falls somewhere between 200 and 230 milliseconds’ 12 which means that there will be an error of approximately 0.215s. To reduce the error in the time for one complete oscillation (i.e. the time period), more oscillations must be timed. By timing ten oscillations, the error in the time period is reduced by a factor of 10; by timing 20 oscillations - the error is reduced by a factor of 20, etc.

Times were recorded for different numbers of oscillations at 0.45m from the centre of gravity:

These results show that 5 oscillations is too few and that 20 is at the other extreme. I decided to time 15 oscillations at the different lengths to reduce the human reaction time error without increasing how long the experiment would last dramatically.

These lengths give a sufficient range of results and I shall use each one in my main experiment. This will give 9 different lengths of the pendulum which will improve the accuracy of the curve and the line of best fit on the straight-line graph.

To ensure that the experiment is a fair test throughout, the equipment will remain unchanged during the entirety of this investigation.

Equipment:

• Digital stop-clock with accuracy of 0.01s - for accurate timing
• Retort stand and clamp
• G-clamp - to prevent the unnecessary movement of the retort stand
• Metre rule - to act as the compound pendulum
• Two corks - 1. for safety; 2. to ensure that the pin is securely positioned
• Blu-tack - for moving the centre of gravity
• Spirit level - to ensure that the pin is horizontal
• Piece of card with a 40º angle marked as a starting line - to ensure that the pendulum is released at the same point

Equipment diagram:

Side view of pivot:

Method:

1. Use the G-clamp to secure the retort stand to the bench.
2. Push the pin through the centre of the cork and out through the other side.
3. Clamp the cork securely with the exposed end forward.
4. Ensure the pin is horizontal by testing it with a spirit level.
5. Suspend the metre rule on the pin at 50cm. If it is in equilibrium, i.e. not moving, skip to 7.
6. Add blu-tack to one end of the metre rule until it balances horizontally without moving. The centre of gravity of the metre rule is now at 50cm along its length.
7. Suspend the metre rule on the pin at 0.05m from the centre of gravity (i.e. at 45 cm along the metre rule)
8. Cover the exposed end of the pin with another cork. Do not push all the way through.
9. Attach the card to the stand with blu-tack ensuring that the vertical timing line is in line with the retort stand.
10. Displace the metre rule so that it lines up with the starting line on the card.
11. Release the pendulum and begin timing as it passes the vertical retort stand.
12. Stop the timer after 15 oscillations as the pendulum passes the retort stand. (One complete oscillation is from when the pendulum passes the stand to it’s extreme of motion at one side, back past the stand to the other extreme and then back to the stand [See Diagram B (above)])
13. Record the time.
14. Repeat steps 10 - 13 for the same length twice. Carry out additional repeats if any values are more than 0.15s from the others.
15. Repeat steps 10 - 14 for lengths of: 0.1m, 0.15m. 0.2m, 0.25m, 0.3m, 0.35m, 0.4m, and 0.45m from the centre of gravity.
16. For each length of the pendulum, h:

• Calculate the average time for 15 oscillations (addition of the values then division by 3)
• Divide the average time by 15 to give the time period
• Calculate T²h
• Calculate h²

And tabulate the results with these calculated values.

17. Plot a graph of T²h against h².